Solve the System of Equations 4x^2+25y^2=100 y=x+2

$4 x^{2} + 25 y^{2} = 100$ $y = x + 2$

Replace all occurrences of $y$ with $x + 2$ in each equation.

Replace all occurrences of $y$ in $4 x^{2} + 25 y^{2} = 100$ with $x + 2$ .

$4 x^{2} + 25 {(x + 2)}^{2} = 100$

$y = x + 2$

Simplify the left side.

Simplify $4 x^{2} + 25 {(x + 2)}^{2}$ .

Simplify each term.

Rewrite ${(x + 2)}^{2}$ as $(x + 2) (x + 2)$ .

$4 x^{2} + 25 ((x + 2) (x + 2)) = 100$

$y = x + 2$

Expand $(x + 2) (x + 2)$ using the FOIL Method.

Apply the distributive property.

$4 x^{2} + 25 (x (x + 2) + 2 (x + 2)) = 100$

$y = x + 2$

Apply the distributive property.

$4 x^{2} + 25 (x \cdot x + x \cdot 2 + 2 (x + 2)) = 100$

$y = x + 2$

Apply the distributive property.

$4 x^{2} + 25 (x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2) = 100$

$y = x + 2$

$4 x^{2} + 25 (x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2) = 100$

$y = x + 2$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$4 x^{2} + 25 (x^{2} + x \cdot 2 + 2 x + 2 \cdot 2) = 100$

$y = x + 2$

Move $2$ to the left of $x$ .

$4 x^{2} + 25 (x^{2} + 2 \cdot x + 2 x + 2 \cdot 2) = 100$

$y = x + 2$

Multiply $2$ by $2$ .

$4 x^{2} + 25 (x^{2} + 2 x + 2 x + 4) = 100$

$y = x + 2$

$4 x^{2} + 25 (x^{2} + 2 x + 2 x + 4) = 100$

$y = x + 2$

Add $2 x$ and $2 x$ .

$4 x^{2} + 25 (x^{2} + 4 x + 4) = 100$

$y = x + 2$

$4 x^{2} + 25 (x^{2} + 4 x + 4) = 100$

$y = x + 2$

Apply the distributive property.

$4 x^{2} + 25 x^{2} + 25 (4 x) + 25 \cdot 4 = 100$

$y = x + 2$

Simplify.

Multiply $4$ by $25$ .

$4 x^{2} + 25 x^{2} + 100 x + 25 \cdot 4 = 100$

$y = x + 2$

Multiply $25$ by $4$ .

$4 x^{2} + 25 x^{2} + 100 x + 100 = 100$

$y = x + 2$

$4 x^{2} + 25 x^{2} + 100 x + 100 = 100$

$y = x + 2$

$4 x^{2} + 25 x^{2} + 100 x + 100 = 100$

$y = x + 2$

Add $4 x^{2}$ and $25 x^{2}$ .

$29 x^{2} + 100 x + 100 = 100$

$y = x + 2$

$29 x^{2} + 100 x + 100 = 100$

$y = x + 2$

$29 x^{2} + 100 x + 100 = 100$

$y = x + 2$

$29 x^{2} + 100 x + 100 = 100$

$y = x + 2$

Solve for $x$ in $29 x^{2} + 100 x + 100 = 100$ .

Subtract $100$ from both sides of the equation.

$29 x^{2} + 100 x + 100 - 100 = 0$

$y = x + 2$

Combine the opposite terms in $29 x^{2} + 100 x + 100 - 100$ .

Subtract $100$ from $100$ .

$29 x^{2} + 100 x + 0 = 0$

$y = x + 2$

Add $29 x^{2} + 100 x$ and $0$ .

$29 x^{2} + 100 x = 0$

$y = x + 2$

$29 x^{2} + 100 x = 0$

$y = x + 2$

Factor $x$ out of $29 x^{2} + 100 x$ .

Factor $x$ out of $29 x^{2}$ .

$x (29 x) + 100 x = 0$

$y = x + 2$

Factor $x$ out of $100 x$ .

$x (29 x) + x \cdot 100 = 0$

$y = x + 2$

Factor $x$ out of $x (29 x) + x \cdot 100$ .

$x (29 x + 100) = 0$

$y = x + 2$

$x (29 x + 100) = 0$

$y = x + 2$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$29 x + 100 = 0$

$y = x + 2$

Set $x$ equal to $0$ .

$x = 0$

$y = x + 2$

Set $29 x + 100$ equal to $0$ and solve for $x$ .

Set $29 x + 100$ equal to $0$ .

$29 x + 100 = 0$

$y = x + 2$

Solve $29 x + 100 = 0$ for $x$ .

Subtract $100$ from both sides of the equation.

$29 x = - 100$

$y = x + 2$

Divide each term in $29 x = - 100$ by $29$ and simplify.

Divide each term in $29 x = - 100$ by $29$ .

$\frac{29 x}{29} = \frac{- 100}{29}$

$y = x + 2$

Simplify the left side.

Cancel the common factor of $29$ .

Cancel the common factor.

$\frac{29 x}{29} = \frac{- 100}{29}$

$y = x + 2$

Divide $x$ by $1$ .

$x = \frac{- 100}{29}$

$y = x + 2$

$x = \frac{- 100}{29}$

$y = x + 2$

$x = \frac{- 100}{29}$

$y = x + 2$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{100}{29}$

$y = x + 2$

$x = - \frac{100}{29}$

$y = x + 2$

$x = - \frac{100}{29}$

$y = x + 2$

$x = - \frac{100}{29}$

$y = x + 2$

$x = - \frac{100}{29}$

$y = x + 2$

The final solution is all the values that make $x (29 x + 100) = 0$ true.

$x = 0, - \frac{100}{29}$

$y = x + 2$

$x = 0, - \frac{100}{29}$

$y = x + 2$

Replace all occurrences of $x$ with $0$ in each equation.

Replace all occurrences of $x$ in $y = x + 2$ with $0$ .

$y = (0) + 2$

$x = 0$

Simplify the right side.

Add $0$ and $2$ .

$y = 2$

$x = 0$

$y = 2$

$x = 0$

$y = 2$

$x = 0$

Replace all occurrences of $x$ with $- \frac{100}{29}$ in each equation.

Replace all occurrences of $x$ in $y = x + 2$ with $- \frac{100}{29}$ .

$y = (- \frac{100}{29}) + 2$

$x = - \frac{100}{29}$

Simplify the right side.

Simplify $(- \frac{100}{29}) + 2$ .

To write $2$ as a fraction with a common denominator, multiply by $\frac{29}{29}$ .

$y = - \frac{100}{29} + 2 \cdot \frac{29}{29}$

$x = - \frac{100}{29}$

Combine $2$ and $\frac{29}{29}$ .

$y = - \frac{100}{29} + \frac{2 \cdot 29}{29}$

$x = - \frac{100}{29}$

Combine the numerators over the common denominator.

$y = \frac{- 100 + 2 \cdot 29}{29}$

$x = - \frac{100}{29}$

Simplify the numerator.

Multiply $2$ by $29$ .

$y = \frac{- 100 + 58}{29}$

$x = - \frac{100}{29}$

Add $- 100$ and $58$ .

$y = \frac{- 42}{29}$

$x = - \frac{100}{29}$

$y = \frac{- 42}{29}$

$x = - \frac{100}{29}$

Move the negative in front of the fraction.

$y = - \frac{42}{29}$

$x = - \frac{100}{29}$

$y = - \frac{42}{29}$

$x = - \frac{100}{29}$

$y = - \frac{42}{29}$

$x = - \frac{100}{29}$

$y = - \frac{42}{29}$

$x = - \frac{100}{29}$

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(0, 2)$

$(- \frac{100}{29}, - \frac{42}{29})$

The result can be shown in multiple forms.

Point Form:

$(0, 2), (- \frac{100}{29}, - \frac{42}{29})$

Equation Form:

$x = 0, y = 2$

$x = - \frac{100}{29}, y = - \frac{42}{29}$

Solve the System of Equations 4x^2+25y^2=100 y=x+2

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